Perturbation of normal quaternionic operators

TL;DR

This paper develops a perturbation theory for quaternionic normal operators in Hilbert spaces, utilizing the $S$-spectrum and slice hyperholomorphicity, addressing unique challenges from non-commutativity.

Contribution

It introduces a novel perturbation framework for quaternionic normal operators using $S$-spectrum and Schatten class operators, extending classical theories.

Findings

Established perturbation results for quaternionic normal operators.

Provided conditions for the existence of hyperinvariant subspaces.

Extended spectral theory to quaternionic operator setting.

Abstract

The theory of quaternionic operators has applications in several different fields such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of spectrum. In fact, in quaternionic operator theory the classical notion of resolvent operator and the one of spectrum need to be replaced by the two $S$-resolvent operators and the $S$-spectrum. This is a consequence of the non-commutativity of the quaternionic setting. Indeed, the $S$-spectrum of a quaternionic linear operator $T$ is given by the non invertibility of a second order operator. This presents new challenges which makes our approach to perturbation theory of quaternionic operators different from the classical case. In this paper we study the problem of perturbation of a quaternionic normal operator in a Hilbert space by making use of the concepts of $S$-spectrum and of slice hyperholomorphicity of the $S$-resolvent operators. For this new setting we prove results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and give conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator.